Towards Globally Convergent Blind Equalization of Constant Modulus Signals: A Bilinear Approach

We consider the problem of blind equalization of a constant modulus signal. One of the most popular classes of algorithms in this context is the Godard family of blind equalizers [1], which includes among others the Constant Modulus Algorithm (CMA) [2]. A common drawback of these algorithms is that they may converge to undesired equalizer settings if not properly initialized. This is known as the problem of ill convergence and is primarily due to the non-convex form of the cost function of algorithms of this class with respect to the equalizer parameters. We propose a di erent approach to the problem, namely, a bilinear one, which leads to a di erent parameterization and to the construction of a convex cost function with respect to the parameters introduced. In a perfectly parameterized case (the equalizer's order matches exactly the order of the channel inverse), the solution to the problem is unique and permits for a direct calculation of the optimal equalizer. In over-parameterized cases however, there exist multiple solutions to our cost function. However, we propose a method that still allows to determine the channel inverse in this case. Di erent adaptive schemes are proposed to adaptively compute the solution of our criterion and the in uence of additive noise is also discussed. 1. The Constant Modulus Algorithm The classical setup for adaptive blind-equalization is shown in gure 1. We denote by ak, xk and yk the transmitted symbol, received sample and equalizer output, respectively, all at time instant k. The equalizer (a FIR lter of order N 1) at the same time instant is denoted by a N 1 vector Wk = [w0 wN 1] T and its output can be written as yk = X H k Wk, where Xk = [xk xk 1 xk N+1] H , T denotes transpose and H complex conjugate transpose. The Godard algorithms are stochastic gradi-